DESCRIPTIVE SET-THEORETIC DICHOTOMY THEOREMS AND LIMITS SUPERIOR C.T. CONLEY, D. LECOMTE, AND B.D. MILLER Abstract. Suppose that X is a Hausdorff space, I is an ideal on X, and (Ai)i2! is a sequence of analytic subsets of X. We inves- tigate the circumstances under which there exists I 2 [!]! with T i2I Ai 2= I. We focus on Laczkovich-style characterizations and ideals associated with descriptive set-theoretic dichotomies. Warning. This is a preliminary draft of our paper. Several argu- ments still need to be written out, there are no doubt many typos remaining, and our terminology is less than perfect. Please email corrections and suggestions to [email protected]. 1. Introduction Given an infinite set I ⊆ !, we use [I]! to denote the family of infinite subsets of I. The limit superior of a sequence (Ai)i2I is given T S 1 by limsupi2I Ai = i2! j2Ini Aj = fx j 9 i 2 I (x 2 Ai)g. Suppose that X is a Hausdorff space. A set A ⊆ X is analytic if it is the continuous image of a closed subset of !!. A set B ⊆ X is Borel if it is in the σ-algebra generated by the topology of X. Given a pointclass Γ of subsets of Hausdorff spaces, we say that an ideal I on X has the Γ limsup property if for all sequences (Bi)i2! ! of subsets of X in Γ, there exists I 2 [!] with limsupi2I Bi 2 I or T B 2= I. Note that if X is analytic, then every Borel subset of X i2I i DRAFT is analytic, so if I has the analytic limsup property, then I has the Borel limsup property. Given a sequence (Xi)i2I of subsets of X, let I (Xi)i2I denote the ideal consisting of all sets Y ⊆ X with the property that 8J 2 [I]!9K 2 ! [J] (limsupk2K Xk \ Y 2 I). A straightforward diagonalization shows that if I is a σ-ideal, then so too is I (Xi)i2I . It is easy to check that I has the Γ limsup property if and only if for all sequences (Bi)i2! ! T of subsets of X in Γ, the existence of I 2 [!] with i2I Bi 2= I is governed by whether X2 = I (Bi)i2!. 1 2 C.T. CONLEY, D. LECOMTE, AND B.D. MILLER As part of the original foray into these notions, Laczkovich showed that the ideal of countable subsets of an uncountable Polish space has the Borel limsup property [13]. Komj´athlater proved that such ideals have the analytic limsup property [12]. Building on subsequent work of Balcerzak-G l¸ab[1], Gao-Jackson-Kieftenbeld have recently established the more general fact that for all co-analytic equivalence relations on Polish spaces, the ideal of sets which intersect only countably many equivalence classes has the analytic limsup property [4]. These are perhaps the best known examples of ideals associated with descriptive set-theoretic dichotomy theorems. It is therefore quite nat- ural to investigate the family of descriptive set-theoretic dichotomy theorems whose associated ideals have the analytic limsup property. Of course, the sheer abundance of such theorems makes this a rather daunting task. Fortunately, recent work [15, 16, 17, 18] indicates that many descrip- tive set-theoretic dichotomy theorems are consequences of a handful of dichotomy theorems concerning chromatic numbers of definable graphs. In particular, many Silver-style dichotomy theorems can be obtained from the Kechris-Solecki-Todorcevic characterization of the class of an- alytic graphs with countable Borel chromatic number [11]. In x2, we give a classical proof that ideals arising from a natural spe- cial case of the Kechris-Solecki-Todorcevic dichotomy theorem [11] have the analytic limsup property. Using this, we give a classical proof of the Gao-Jackson-Kieftenbeld theorem [4], answering a question of Gao. We also prove that ideals associated with Feng's special case of the open coloring axiom [3], the Friedman-Harrington-Kechris characteri- zation of separable quasi-metric spaces [10], van Engelen-Kunen-Miller- style characterizations of vector spaces which are unions of countably many low-dimensional subspaces [2], and the Friedman-Shelah charac- terization of separable linear quasi-orders [19] have the analytic limsup property. Generalizing a result of Balcerzak-G l¸ab[1], we show that products of these ideals with analytically principal ideals have the an- alytic limsupDRAFT property. Generalizing results of Balcerzak-G l¸ab[1] and Gao-Jackson-Kieftenbeld [4], we show that these ideals satisfy a para- metric strengthening of the analytic limsup property. We also discuss generalizations to κ-Souslin structures. In x3, we establish that non-trivial ideals arising from the locally countable special case of the Kechris-Solecki-Todorcevic dichotomy the- orem [11] do not have the compact limsup property. Using this, we show that non-trivial ideals associated with the Harrington-Kechris- Louveau dichotomy theorem [6] do not have the compact limsup prop- erty, answering another question of Gao. We also characterize the DICHOTOMY THEOREMS AND LIMITS SUPERIOR 3 ideals associated with the Lusin-Novikov uniformization theorem (see x18 of [9]) which have the analytic limsup property, and we show that products of ideals which have analytic perfect antichains with analyt- ically non-principal ideals do not have the analytic limsup property. This implies that products of non-trivial ideals associated with de- scriptive set-theoretic dichotomy theorems do not have the analytic limsup property, and negatively answers Balcerzak-G l¸ab'squestion [1] as to whether the product of the trivial ideal on an uncountable Po- lish space with the ideal of countable subsets of an uncountable Polish space has the analytic limsup property, which gives rise to ideals asso- ciated with the Harrington-Marker-Shelah Borel-Dilworth theorem [7] which do not have the analytic limsup property. 2. Positive results A graph on X is an irreflexive symmetric set G ⊆ X × X. The restriction of G to a set A ⊆ X is given by G A = G \ (A × A). We say that A is G-discrete if G A = ;. It is a well-known corollary of the first separation theorem that if G is analytic, then every G-discrete analytic set is contained in a G-discrete Borel set. We say that a graph G on X has the limsup property if for all se- S quences (Ai)i2! of subsets of X, sets S, s 2 S, and R ⊆ X , there ! S exists I 2 [!] such that projs(R \ limsupi2I Ai ) is G-discrete or T S projs(R \ i2I Ai ) is not G-discrete. Proposition 1. Suppose that X is a set and G is a graph on X with transitive complement. Then G has the limsup property. Proof. Suppose that (Ai)i2! is a sequence of subsets of X, S is a set, S S s 2 S, R ⊆ X , and projs(R \ limsupi2I Ai ) is not G-discrete for all ! S ! T S I 2 [!] . Fix x 2 R \ limsupi2! Ai , I 2 [!] with x 2 i2I Ai , and S y; z 2 R \ limsupi2I Ai with (y(s); z(s)) 2 G. As the complement of G is transitive, it follows that (w(s); x(s)) 2 G for some w 2 fy; zg. ! T S T S Fix J 2 [I] with w 2 j2J Aj , and note that w; x 2 R \ j2J Aj and DRAFTT S (w(s); x(s)) 2 G, so projs(R \ j2J Aj ) is not G-discrete, thus G has the limsup property. Proposition 2. Suppose that X is a set and G is a graph on X which can be written as the union of countably many rectangles. Then G has the limsup property. S Proof. Fix sets Bk;Ck ⊆ X such that G = k2! Bk × Ck, and suppose that (Ai)i2! is a sequence of subsets of X, S is a set, s 2 S, and S ! R ⊆ X . We will recursively construct sets Ik 2 [!] for k 2 !, 4 C.T. CONLEY, D. LECOMTE, AND B.D. MILLER beginning with I = !. If B \ proj (R \ limsup AS) = ;, then 0 k s i2Ik i we set I = I . Otherwise, we fix x 2 R \ limsup AS with k+1 k k i2Ik i x (s) 2 B , as well as I 2 [I ]! such that x 2 T AS. k k k+1 k k i2Ik+1 i ! Fix I 2 [!] with jI n Ikj < @0 for all k 2 !, and suppose that S projs(R \ limsupi2I Ai ) is not G-discrete. Then there exist x; y 2 S R \ limsupi2I Ai with (x(s); y(s)) 2 G and k 2 ! with (x(s); y(s)) 2 ! T S Bk × Ck, so xk is defined. Fix J 2 [Ik+1] with y 2 j2J Aj . Then T S T S xk; y 2 R \ j2J Aj and (xk(s); y(s)) 2 G, so projs(R \ j2J Aj ) is not G-discrete, thus G has the limsup property. A Y -coloring of G is a function c: X ! Y which sends G-related points of X to distinct points of Y . More generally, a homomorphism from a graph G on X to a graph H on Y is a function π : X ! Y which sends G-related points of X to H-related points of Y . We use IG to denote the σ-ideal generated by the family of G-discrete Borel subsets of X. It is easy to see that X 2 IG if and only if there is a Borel !-coloring of G.